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G = C24.462C23order 128 = 27

302nd non-split extension by C24 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C24.462C23, C23.717C24, C22.4902+ 1+4, C22.3742- 1+4, (C22×C4).412D4, C23.227(C2×D4), C2.64(C233D4), (C22×C4).228C23, (C23×C4).502C22, C22.449(C22×D4), C23.34D4.35C2, C23.11D4.59C2, (C22×Q8).233C22, C23.78C2363C2, C23.83C23131C2, C2.C42.420C22, C2.68(C23.38C23), C2.58(C22.57C24), (C2×C4).434(C2×D4), (C2×C22⋊Q8).48C2, (C2×C4⋊C4).526C22, (C2×C22⋊C4).336C22, SmallGroup(128,1549)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.462C23
C1C2C22C23C22×C4C22×Q8C23.78C23 — C24.462C23
C1C23 — C24.462C23
C1C23 — C24.462C23
C1C23 — C24.462C23

Generators and relations for C24.462C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=f2=g2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, geg-1=abe >

Subgroups: 436 in 222 conjugacy classes, 92 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×16], C22, C22 [×6], C22 [×10], C2×C4 [×4], C2×C4 [×44], Q8 [×8], C23, C23 [×2], C23 [×6], C22⋊C4 [×10], C4⋊C4 [×16], C22×C4 [×18], C22×C4 [×2], C2×Q8 [×8], C24, C2.C42 [×16], C2×C22⋊C4 [×6], C2×C4⋊C4 [×10], C22⋊Q8 [×8], C23×C4, C22×Q8 [×2], C23.34D4, C23.78C23 [×4], C23.11D4 [×4], C23.83C23 [×4], C2×C22⋊Q8 [×2], C24.462C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], 2- 1+4 [×4], C233D4, C23.38C23 [×2], C22.57C24 [×4], C24.462C23

Character table of C24.462C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-1-11-11111-1-11-1-1    linear of order 2
ρ311111111111111-1-1111-1-1-11-1-1-1    linear of order 2
ρ41111111111-1-1-1-11-1-111-1-11-1-111    linear of order 2
ρ511111111-1-11-11-11-111-1111-1-1-1-1    linear of order 2
ρ611111111-1-1-11-11-1-1-11-111-11-111    linear of order 2
ρ711111111-1-11-11-1-1111-1-1-1-1-1111    linear of order 2
ρ811111111-1-1-11-1111-11-1-1-1111-1-1    linear of order 2
ρ91111111111-1-1-1-1111-1-1-11-11-11-1    linear of order 2
ρ1011111111111111-11-1-1-1-111-1-1-11    linear of order 2
ρ111111111111-1-1-1-1-1-11-1-11-1111-11    linear of order 2
ρ12111111111111111-1-1-1-11-1-1-111-1    linear of order 2
ρ1311111111-1-1-11-111-11-11-11-1-11-11    linear of order 2
ρ1411111111-1-11-11-1-1-1-1-11-111111-1    linear of order 2
ρ1511111111-1-1-11-11-111-111-11-1-11-1    linear of order 2
ρ1611111111-1-11-11-111-1-111-1-11-1-11    linear of order 2
ρ172-22-22-22-22-2-222-2000000000000    orthogonal lifted from D4
ρ182-22-22-22-2-22-2-222000000000000    orthogonal lifted from D4
ρ192-22-22-22-22-22-2-22000000000000    orthogonal lifted from D4
ρ202-22-22-22-2-2222-2-2000000000000    orthogonal lifted from D4
ρ214-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-444-4-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ24444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C24.462C23
On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 23)(2 24)(3 21)(4 22)(5 11)(6 12)(7 9)(8 10)(13 64)(14 61)(15 62)(16 63)(17 25)(18 26)(19 27)(20 28)(29 33)(30 34)(31 35)(32 36)(37 43)(38 44)(39 41)(40 42)(45 51)(46 52)(47 49)(48 50)(53 57)(54 58)(55 59)(56 60)
(1 27)(2 28)(3 25)(4 26)(5 61)(6 62)(7 63)(8 64)(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)(29 40)(30 37)(31 38)(32 39)(33 42)(34 43)(35 44)(36 41)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 47 3 45)(2 46 4 48)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 32 15 30)(14 31 16 29)(17 53 19 55)(18 56 20 54)(21 51 23 49)(22 50 24 52)(25 57 27 59)(26 60 28 58)(33 61 35 63)(34 64 36 62)
(1 32 3 30)(2 33 4 35)(5 48 7 46)(6 51 8 49)(9 52 11 50)(10 47 12 45)(13 59 15 57)(14 56 16 54)(17 43 19 41)(18 38 20 40)(21 34 23 36)(22 31 24 29)(25 37 27 39)(26 44 28 42)(53 64 55 62)(58 61 60 63)
(1 4 3 2)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 20 19 18)(21 24 23 22)(25 28 27 26)(29 34 31 36)(30 35 32 33)(37 44 39 42)(38 41 40 43)(45 50 47 52)(46 51 48 49)(53 60 55 58)(54 57 56 59)(61 62 63 64)

G:=sub<Sym(64)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,23)(2,24)(3,21)(4,22)(5,11)(6,12)(7,9)(8,10)(13,64)(14,61)(15,62)(16,63)(17,25)(18,26)(19,27)(20,28)(29,33)(30,34)(31,35)(32,36)(37,43)(38,44)(39,41)(40,42)(45,51)(46,52)(47,49)(48,50)(53,57)(54,58)(55,59)(56,60), (1,27)(2,28)(3,25)(4,26)(5,61)(6,62)(7,63)(8,64)(9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24)(29,40)(30,37)(31,38)(32,39)(33,42)(34,43)(35,44)(36,41)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,47,3,45)(2,46,4,48)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,32,15,30)(14,31,16,29)(17,53,19,55)(18,56,20,54)(21,51,23,49)(22,50,24,52)(25,57,27,59)(26,60,28,58)(33,61,35,63)(34,64,36,62), (1,32,3,30)(2,33,4,35)(5,48,7,46)(6,51,8,49)(9,52,11,50)(10,47,12,45)(13,59,15,57)(14,56,16,54)(17,43,19,41)(18,38,20,40)(21,34,23,36)(22,31,24,29)(25,37,27,39)(26,44,28,42)(53,64,55,62)(58,61,60,63), (1,4,3,2)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,20,19,18)(21,24,23,22)(25,28,27,26)(29,34,31,36)(30,35,32,33)(37,44,39,42)(38,41,40,43)(45,50,47,52)(46,51,48,49)(53,60,55,58)(54,57,56,59)(61,62,63,64)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,23)(2,24)(3,21)(4,22)(5,11)(6,12)(7,9)(8,10)(13,64)(14,61)(15,62)(16,63)(17,25)(18,26)(19,27)(20,28)(29,33)(30,34)(31,35)(32,36)(37,43)(38,44)(39,41)(40,42)(45,51)(46,52)(47,49)(48,50)(53,57)(54,58)(55,59)(56,60), (1,27)(2,28)(3,25)(4,26)(5,61)(6,62)(7,63)(8,64)(9,16)(10,13)(11,14)(12,15)(17,21)(18,22)(19,23)(20,24)(29,40)(30,37)(31,38)(32,39)(33,42)(34,43)(35,44)(36,41)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,47,3,45)(2,46,4,48)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,32,15,30)(14,31,16,29)(17,53,19,55)(18,56,20,54)(21,51,23,49)(22,50,24,52)(25,57,27,59)(26,60,28,58)(33,61,35,63)(34,64,36,62), (1,32,3,30)(2,33,4,35)(5,48,7,46)(6,51,8,49)(9,52,11,50)(10,47,12,45)(13,59,15,57)(14,56,16,54)(17,43,19,41)(18,38,20,40)(21,34,23,36)(22,31,24,29)(25,37,27,39)(26,44,28,42)(53,64,55,62)(58,61,60,63), (1,4,3,2)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,20,19,18)(21,24,23,22)(25,28,27,26)(29,34,31,36)(30,35,32,33)(37,44,39,42)(38,41,40,43)(45,50,47,52)(46,51,48,49)(53,60,55,58)(54,57,56,59)(61,62,63,64) );

G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,23),(2,24),(3,21),(4,22),(5,11),(6,12),(7,9),(8,10),(13,64),(14,61),(15,62),(16,63),(17,25),(18,26),(19,27),(20,28),(29,33),(30,34),(31,35),(32,36),(37,43),(38,44),(39,41),(40,42),(45,51),(46,52),(47,49),(48,50),(53,57),(54,58),(55,59),(56,60)], [(1,27),(2,28),(3,25),(4,26),(5,61),(6,62),(7,63),(8,64),(9,16),(10,13),(11,14),(12,15),(17,21),(18,22),(19,23),(20,24),(29,40),(30,37),(31,38),(32,39),(33,42),(34,43),(35,44),(36,41),(45,57),(46,58),(47,59),(48,60),(49,55),(50,56),(51,53),(52,54)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,47,3,45),(2,46,4,48),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,32,15,30),(14,31,16,29),(17,53,19,55),(18,56,20,54),(21,51,23,49),(22,50,24,52),(25,57,27,59),(26,60,28,58),(33,61,35,63),(34,64,36,62)], [(1,32,3,30),(2,33,4,35),(5,48,7,46),(6,51,8,49),(9,52,11,50),(10,47,12,45),(13,59,15,57),(14,56,16,54),(17,43,19,41),(18,38,20,40),(21,34,23,36),(22,31,24,29),(25,37,27,39),(26,44,28,42),(53,64,55,62),(58,61,60,63)], [(1,4,3,2),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,20,19,18),(21,24,23,22),(25,28,27,26),(29,34,31,36),(30,35,32,33),(37,44,39,42),(38,41,40,43),(45,50,47,52),(46,51,48,49),(53,60,55,58),(54,57,56,59),(61,62,63,64)])

Matrix representation of C24.462C23 in GL10(𝔽5)

1000000000
0100000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000004000
0000000400
0000000040
0000000004
,
4000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000001000
0000000100
0000000010
0000000001
,
4000000000
0400000000
0042000000
0041000000
0000420000
0000410000
0000004000
0000000100
0000000440
0000001001
,
0300000000
2000000000
0020000000
0023000000
0000300000
0000320000
0000000430
0000004003
0000000001
0000000010
,
0100000000
1000000000
0000100000
0000010000
0040000000
0004000000
0000000100
0000001000
0000000001
0000000010
,
1000000000
0100000000
0013000000
0014000000
0000420000
0000410000
0000004000
0000000400
0000000110
0000001001

G:=sub<GL(10,GF(5))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1],[0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,3,1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1] >;

C24.462C23 in GAP, Magma, Sage, TeX

C_2^4._{462}C_2^3
% in TeX

G:=Group("C2^4.462C2^3");
// GroupNames label

G:=SmallGroup(128,1549);
// by ID

G=gap.SmallGroup(128,1549);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,794,185,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=f^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,g*e*g^-1=a*b*e>;
// generators/relations

Export

Character table of C24.462C23 in TeX

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